Two letters are chosen at random without replacement from the word HOME. What is the probability that none of the chosen letters is a vowel?

Introduction / Context:
This question involves probability with letters from a short word. The aim is to draw two letters such that neither is a vowel. It is a simple combination problem since the order of drawing letters is not important, and the letters are drawn without replacement.

Given Data / Assumptions:

• The word HOME has 4 letters: H, O, M and E.
• Vowels in HOME: O and E (2 vowels).
• Consonants in HOME: H and M (2 consonants).
• Two letters are chosen at random without replacement.
• Event of interest: both chosen letters are consonants.

Concept / Approach:
Order does not matter when simply choosing letters, so we use combinations. The probability of selecting two consonants is equal to the number of ways to choose 2 consonants divided by the total number of ways to choose any 2 letters from the 4 available letters.

Step-by-Step Solution:
Total number of letters = 4. Total ways to choose 2 letters = C(4, 2) = 4 * 3 / 2 = 6. Number of consonants = 2 (H and M). Ways to choose 2 consonants from 2 = C(2, 2) = 1. Probability that both letters are consonants = favourable / total = 1 / 6.

Verification / Alternative check:
We can list all possible pairs of letters: HO, HM, HE, OM, OE, ME. Only HM consists of two consonants. There are 6 pairs total and 1 favourable pair, so the probability is 1/6, confirming the combination approach.

Why Other Options Are Wrong:
1/2, 1/3 and 1/4 would imply 3, 2, or 1.5 favourable outcomes, which do not match the actual single favourable pair. Thus any fraction other than 1/6 misrepresents the counting of consonant pairs.

Common Pitfalls:
A common mistake is to forget that HOME has two vowels and two consonants and to miscount consonants or vowels. Another error is to think order matters and use permutations, which leads to an incorrect denominator. Since the question does not care about order, combinations are the correct tool.

Final Answer:
Hence, the probability that neither of the chosen letters is a vowel is 1/6.